In this Letter, we show that, at low temperatures, zero-point fluctuations of the plasmon modes of two mutually coupled 2D planar Wigner crystals give rise to a novel long-range attractive force. For the case where the distance d between two planar surfaces is large, this attractive force has an unusual power-law decay, which scales as d(-7/2), unlike other fluctuation-induced forces. Specifically, we note that its range is longer than the "standard" zero-temperature van der Waals interaction. This result may, in principle, be observed in bilayer electronic systems and provides insight into the nature of correlation effects for highly charged surfaces.